Optimal. Leaf size=52 \[ \frac {\sqrt {1-2 x^2} \sqrt {1-x^2} F\left (\left .\sin ^{-1}(x)\right |2\right )}{\sqrt {x-1} \sqrt {x+1} \sqrt {2 x^2-1}} \]
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Rubi [A] time = 0.05, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {519, 421, 419} \[ \frac {\sqrt {1-2 x^2} \sqrt {1-x^2} F\left (\left .\sin ^{-1}(x)\right |2\right )}{\sqrt {x-1} \sqrt {x+1} \sqrt {2 x^2-1}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 421
Rule 519
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+x} \sqrt {1+x} \sqrt {-1+2 x^2}} \, dx &=\frac {\sqrt {-1+x^2} \int \frac {1}{\sqrt {-1+x^2} \sqrt {-1+2 x^2}} \, dx}{\sqrt {-1+x} \sqrt {1+x}}\\ &=\frac {\left (\sqrt {1-2 x^2} \sqrt {-1+x^2}\right ) \int \frac {1}{\sqrt {1-2 x^2} \sqrt {-1+x^2}} \, dx}{\sqrt {-1+x} \sqrt {1+x} \sqrt {-1+2 x^2}}\\ &=\frac {\left (\sqrt {1-2 x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {1-2 x^2} \sqrt {1-x^2}} \, dx}{\sqrt {-1+x} \sqrt {1+x} \sqrt {-1+2 x^2}}\\ &=\frac {\sqrt {1-2 x^2} \sqrt {1-x^2} F\left (\left .\sin ^{-1}(x)\right |2\right )}{\sqrt {-1+x} \sqrt {1+x} \sqrt {-1+2 x^2}}\\ \end {align*}
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Mathematica [B] time = 0.24, size = 107, normalized size = 2.06 \[ -\frac {2 (x-1)^{3/2} \sqrt {\frac {x+1}{1-x}} \sqrt {\frac {1-2 x^2}{(x-1)^2}} F\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt {2}+2+\frac {1}{x-1}}}{2^{3/4}}\right )|4 \left (-4+3 \sqrt {2}\right )\right )}{\sqrt {3+2 \sqrt {2}} \sqrt {x+1} \sqrt {2 x^2-1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {2 \, x^{2} - 1} \sqrt {x + 1} \sqrt {x - 1}}{2 \, x^{4} - 3 \, x^{2} + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 \, x^{2} - 1} \sqrt {x + 1} \sqrt {x - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 58, normalized size = 1.12 \[ \frac {\sqrt {x -1}\, \sqrt {x +1}\, \sqrt {2 x^{2}-1}\, \sqrt {-x^{2}+1}\, \sqrt {-2 x^{2}+1}\, \EllipticF \left (x , \sqrt {2}\right )}{2 x^{4}-3 x^{2}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 \, x^{2} - 1} \sqrt {x + 1} \sqrt {x - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {2\,x^2-1}\,\sqrt {x-1}\,\sqrt {x+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x - 1} \sqrt {x + 1} \sqrt {2 x^{2} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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